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Bloch torsion and modular combinations of Nahm sums
Abstract
We construct a positive-definite symmetric integral 12×12 matrix A such that, for every solution Q of Nahm's equations 1−Q=QA with Qi≠0,1 for all i, the associated element ∑i[Qi] is torsion in the Bloch group of ℚ(Q1,…,Q12). However, the Nahm sum FA,B,C(τ) is nonmodular for every B∈ℚ12 and C∈ℚ. Thus the torsion condition for all solutions in Nahm's conjecture is not sufficient for the existence of a modular Nahm sum.
Related
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BibTeX
@misc{guo2026bloch,
title = {{Bloch torsion and modular combinations of Nahm sums}},
author = {Xuejun Guo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15587},
eprint = {2610.15587},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15587},
}
RIS
TY - UNPB
TI - Bloch torsion and modular combinations of Nahm sums
AU - Xuejun Guo
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15587
UR - https://arxiv.org/abs/2610.15587
AB - We construct a positive-definite symmetric integral 12×12 matrix A such that, for every solution Q of Nahm's equations 1−Q=Q^A with Q_i≠0,1 for all i, the associated element ∑_i[Q_i] is torsion in the Bloch group of ℚ(Q_1,…,Q_12). However, the Nahm sum F_A,B,C(τ) is nonmodular for every B∈ℚ^12 and C∈ℚ. Thus the torsion condition for all solutions in Nahm's conjecture is not sufficient for the existence of a modular Nahm sum.
ER -